Inverse coefficient problems for variational inequalities : optimality conditions and numerical realization
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001) no. 1, pp. 129-152.

We consider the identification of a distributed parameter in an elliptic variational inequality. On the basis of an optimal control problem formulation, the application of a primal-dual penalization technique enables us to prove the existence of multipliers giving a first order characterization of the optimal solution. Concerning the parameter we consider different regularity requirements. For the numerical realization we utilize a complementarity function, which allows us to rewrite the optimality conditions as a set of equalities. Finally, numerical results obtained from a least squares type algorithm emphasize the feasibility of our approach.

Classification : 49N50,  35R30,  35J85
Mots clés : bilevel problem, complementarity function, inverse problem, optimal control, variational inequality
@article{M2AN_2001__35_1_129_0,
author = {Hinterm\"uller, Michael},
title = {Inverse coefficient problems for variational inequalities : optimality conditions and numerical realization},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {129--152},
publisher = {EDP-Sciences},
volume = {35},
number = {1},
year = {2001},
zbl = {0978.65054},
mrnumber = {1811984},
language = {en},
url = {http://www.numdam.org/item/M2AN_2001__35_1_129_0/}
}
Hintermüller, Michael. Inverse coefficient problems for variational inequalities : optimality conditions and numerical realization. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001) no. 1, pp. 129-152. http://www.numdam.org/item/M2AN_2001__35_1_129_0/

[1] V. Barbu, Optimal Control of Variational Inequalities. Res. Notes Math., Pitman, 100 (1984). | MR 742624 | Zbl 0574.49005

[2] B. Bayada and M. El Aalaoui Talibi, Control by the coefficients in a variational inequality: the inverse elastohydrodynamic lubrication problem. Report no. 173, I.N.S.A. Lyon (1994).

[3] A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam (1978). | MR 503330 | Zbl 0404.35001

[4] M. Bergounioux, Optimal control problems governed by abstract elliptic variational inequalities with state constraints. SIAM J. Control Optim. 36 (1998) 273-289. | Zbl 0919.49002

[5] M. Bergounioux and H. Dietrich, Optimal control problems governed by obstacle type variational inequalities: a dual regularization penalization approach. J. Convex Anal. 5 (1998) 329-351. | Zbl 0919.49003

[6] M. Bergounioux, K. Ito and K. Kunisch, Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37 (1999) 1176-1194. | Zbl 0937.49017

[7] M. Bergounioux and F. Mignot, Optimal control of obstacle problems: existence of Lagrange multipliers. ESAIM: COCV 5 (2000) 45-70. | Numdam | Zbl 0934.49008

[8] A. Bermudez and C. Saguez, Optimality conditions for optimal control problems of variational inequalities, in: Control problems for systems described by partial differential equations and applications. I. Lasiecka and R. Triggiani Eds., Lect. Notes Control and Information Sciences, Springer, Berlin (1987). | MR 910512 | Zbl 0627.49011

[9] G. Capriz and G. Cimatti, Free boundary problems in the theory of hydrodynamic lubrication: a survey, in: Free Boundary Problems: Theory and Applications, Vol. II, A. Fasano and M. Primicerio Eds., Res. Notes Math., Pitman, 79 (1983). | MR 714938 | Zbl 0557.76038

[10] G. Cimatti, On a problem of the theory of lubrication governed by a variational inequality. Appl. Math. Optim. 3 (1977) 227-242. | Zbl 0404.76036

[11] F. Clarke, Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983). | MR 709590 | Zbl 0582.49001

[12] J. Dennis and R. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Series in Computational Mathematics, Prentice-Hall, Englewood Cliffs, New Jersey (1983). | MR 702023 | Zbl 0579.65058

[13] F. Facchinei, A. Fischer and C. Kanzow, A semismooth Newton method for variational inequalities: the case of box constraints, in Complementarity and Variational Problems, State of the Art, M. Ferris and J. Pang Eds., SIAM, Philadelphia (1997). | MR 1445073 | Zbl 0886.90152

[14] F. Facchinei, H. Jiang and L. Qi, A smoothing method for mathematical programs with equilibrium constraints. Math. Prog. 85 (1999) 107-134. | Zbl 0959.65079

[15] J. Guo, A variational inequality associated with a lubrication problem, IMA Preprint Series, no. 530 (1989).

[16] B. Hu, A quasi-variational inequality arising in elastohydrodynamics. SIAM J. Math. Anal. 21 (1990) 18-36. | Zbl 0718.35101

[17] K. Ito and K. Kunisch, On the injectivity and linearization of the coefficient-to-solution mapping for elliptic boundary value problems. J. Math. Anal. Appl. 188 (1994) 1040-1066. | Zbl 0817.35021

[18] K. Ito and K. Kunisch, Optimal control of elliptic variational inequalities. Appl. Math. Optim. 41 (2000) 343-364. | Zbl 0960.49003

[19] W. Liu and J. Rubio, Optimality conditions for strongly monotone variational inequalities. Appl. Math. Optim. 27 (1993) 291-312. | Zbl 0779.49012

[20] Z. Luo, J. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints. Cambridge University Press, New York (1996). | MR 1419501 | Zbl 0898.90006

[21] Z. Luo and P. Tseng, A new class of merit functions for the nonlinear complementarity problem, in Complementarity and Variational Problems, State of the Art, M. Ferris and J. Pang Eds., SIAM, Philadelphia (1997). | MR 1445081 | Zbl 0886.90158

[22] F. Mignot and J.P. Puel, Optimal control in some variational inequalities. SIAM J. Control Optim. 22 (1984) 466-476. | Zbl 0561.49007